I know that TeX Users Group recommends putting your local texmf tree at ~/Library/texmf (see link). However, I want MacTeX to see my local texmf tree in a folder in my Dropbox. This allows me to refer to the same local texmf tree on multiple computers, which makes updating my local texmf tree easier (just update the version in the Dropbox and all the computers see the update).

How do I do this?

Note: I found some similar tex.stackexchange questions, but they don't answer my question.

Although this worked, it's not a long-term solution, since there is something wrong with how your path is set up in the command line. On a properly set up MacTeX system, which tlmgr should return /usr/texbin/tlmgr. Do you possibly have a MacPorts version of TeX installed as well? What does echo $PATH return?
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Alan MunnNov 16 '13 at 3:10

@AlanMunn I installed MacTeX by downloading MacTeX.pkg from tug. Should I instead have used MacPorts or Homebrew? echo $PATH returns /usr/local/bin:/usr/bin:/bin:/usr/sbin:/sbin:/usr/local/bin:/usr/X11/bin. According to tug FAQ, the PATH should be set up manually by modifying the shell startup script.
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I Like to CodeNov 16 '13 at 15:10

No, you shouldn't have installed via MacPorts or Homebrew. But it's very odd that your path isn't set correctly. See my answer to Where is PATH modified to include /usr/texbin? for the correct way to fix your path system-wide. You may need to logout and back in for it to take effect. (Or perhaps just close and reopen the Termina.)
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Alan MunnNov 16 '13 at 20:13

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Perhaps you are overwriting the path completely in a .bash_profile or .bashrc? If you make a brand new account, does the path show up correctly?
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Alan MunnNov 17 '13 at 4:11

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@AlanMunn Indeed, there is a .bash_profile file that has the contents export PATH=/usr/local/bin:/usr/bin:/bin:/usr/sbin:/sbin:/usr/local/bin:/usr/X11/bin. When I delete the .bash_profile file, and restart Terminal, now /usr/texbin is on the path and the Terminal can find the tlmgr command. Thanks!
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I Like to CodeNov 17 '13 at 17:43